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Ghosts and Strong Ghosts in the Stable Category

Published online by Cambridge University Press:  20 November 2018

Jon F. Carlson ,
Sunil K. Chebolu  and
Ján Mináč
Jon F. Carlson
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, USA e-mail: jfc@math.uga.edu
Sunil K. Chebolu
Affiliation:
Department of Mathematics, Illinois State University, Normal, IL 61790 USA e-mail: schebol@ilstu.edu
Ján Mináč
Affiliation:
Department of Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada e-mail: minac@uwo.ca
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Abstract

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Suppose that $G$ is a finite group and $k$ is a field of characteristic $p\,>\,0$. A ghost map is a map in the stable category of finitely generated $kG$-modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper we showed that the thick subcategory generated by the trivial module has no nonzero ghost maps if and only if the Sylow $p$-subgroup of $G$ is cyclic of order $2$ or $3$. In this paper we introduce and study variations of ghost maps. In particular, we consider the behavior of ghost maps under restriction and induction functors. We find all groups satisfying a strong form of Freyd’s generating hypothesis and show that ghosts can be detected on a finite range of degrees of Tate cohomology. We also consider maps that mimic ghosts in high degrees.

Keywords

20C2020J0655P42Tate cohomologyghost mapsstable module categoryalmost split sequenceperiodic cohomology
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 4 , 01 December 2016 , pp. 682 - 692
Copyright
Copyright © Canadian Mathematical Society 2016

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